For a countable family of functions \(\mathcal {A}=\lbrace \psi _i: i\in I\rbrace  \subset L^2(X;\mathcal {H}),\)  let the multiplication generated system \( \mathcal {E}_{\mathcal {D}}(\mathcal {A})\)  be Bessel. Then for a.e. \(x\in X\) , the system \(\lbrace \psi _i(x): i\in I\rbrace \)  is Bessel [1]}. The analysis operator \(T_{\mathcal {A}}(x): J_{\mathcal {A}} (x)\rightarrow \ell ^2(I) \)  of \(\lbrace \psi _i(x): i\in I\rbrace \)  is given by
\(T_{\mathcal {A}}(x)(h)= \lbrace \langle h, \psi _i(x) \rangle \rbrace _{i\in I}\,\,\,\, \text{for every $h \in J_{\mathcal {A}}(x)$}.\) 
